A Pythagorean triple is a group of three positive integers like (3, 4, 5), where the sum of the squares of the two smaller integers is equal to the square of the largest integer.

Some Pythagorean triples, for example (3, 4, 5), contain exactly two prime numbers.

Are there any Pythagorean triples that contain

(i)   no prime numbers?

(ii)   one prime number?

(iii)  three prime numbers?

A cuboid measures p cm x q cm by r cm, where p, q and r are all prime. Show that the length of the diagonal of the cuboid cannot be a whole number of cm.

(Remember to give explanations or evidence to support your answers.)